If someone used common wood (oak, hickory, etc) from the woodpile, which had been sitting in the rain and weather for some moderate time and, consequently, may be harboring some common fungi, could the smoke from such wood, deposited on food, cause illness?
The first thing I would suggest is finding a copy of "Food and Beverage Mycology, 2nd. Edition" by Larry R. Beuchat. There are two chapters that are directly pertinent to your question: "field and storage fungi" (p211-232) and "mycotoxins" (p517-570). However, in order to specifically answer the question, one would need to be familiar with three additional chapters: 50 pages of classification of yeast and molds at the beginning of the book, and two equally substantial chapters on detection of mycotoxins and fungi.
Without identifying the exact species of fungus on the wood, and therefore what particular toxins it may produce (many produce toxins that are harmful to us, some produce ones that are only harmful to other fungi), it is impossible to say that it is safe to smoke with this wood. Some of the toxins may be degraded by heat, some may become even more hazardous. Even with all of my equipment in my "kitchen lab" (including an incubator that I use for growing edible fungi), I would just use the mold-covered wood for a nice fire around which my neighbors could gather and share some barbecue smoked on clean, dry wood (scientific studies show that soaking wood does not actually do anything but get the very outside layer wet).